Introduction to Number TheoryChapters4,7,8,9 by William W. Adams, Larry Joel Goldstein

By William W. Adams, Larry Joel Goldstein

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The highest common factor of the two polynomials m(X ) and q(X ) must be a factor of m(X ); as m(X ) is irreducible, its only factors are 1 and m(X ) itself. However, m(X ) is not a factor of q(X ), as m(α) = 0, but q(α) ̸= 0. By a similar argument to the discussion of the Euclidean algorithm in Chap. 6), there are polynomials s(X ) and t (X ) over Q such that s(X )q(X ) + t (X )m(X ) = 1. In particular, s(α)q(α) + t (α)m(α) = 1, and therefore s(α)q(α) = 1, because m(α) = 0. We conclude that 1/q(α) = s(α), and so p(α)/q(α) = p(α)s(α), a polynomial expression in α, as required.

7 Let α be a root of X 4 + 2X + 1 = 0. Write in α with rational coefficients. 4 Number Fields Although A is countable, it is still very much larger than the rational numbers Q (it has infinite degree over Q, for example), and is too large to be really useful. 13 A field K is a number field if it is a finite extension of Q. , the dimension of K as a vector space over Q. 9, is necessarily algebraic. 14 1. Q itself is a number field. Indeed, it will serve as the inspiration for our general theory.

Recall that the degree of the field extension Q(α)/Q is the dimension of the set Q(α) when regarded as a vector space over Q; that is, it is the number of elements in as a a basis {ω1 , . . , ωn } so that every element of Q(α) can be expressed uniquely √ sum a1 ω1 + · · · + an ωn with ai ∈ Q. √For example, Q( 2) has degree 2 over Q, as each element can be written as a + b 2. 4 Show that Q( 2, 3) has degree 4 over Q by proving that 1, 2, 3 and 6 are linearly independent. 9 Let α be a complex number.

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